23 square-pattern orbits and their parametric families.
The open 9/9 problem
Does a 3×3 magic square of nine distinct perfect squares exist?
The required integral square must have nine positive, pairwise distinct perfect-square entries. Neither such a square nor a proof of its impossibility is currently known.
ABCDGa partial 5/9 case
A17689
B27889
C11449
D12769
E19009
F25249
G26569
H10129
J20329
E19 009x−1 320y7 560
5/9 squares
Approach to the problem
Partial configurations
To separate the positional and arithmetic restrictions of the 9/9 problem, the study considers k/9 squares in which only selected entries are required to be squares. Classifications and parametrizations of these patterns do not solve the original problem, but describe its partial cases.
23 square-pattern orbits and general parametrization methods.
An atlas of positional types, tfmn classes, and individual elliptic surfaces.
Neither the required square nor a proof of its impossibility is known.
Results arising from the investigation
Related results
The study of partial configurations led to independent results that are not limited to the 9/9 problem.
Square classes and congruent numbers
The functions f and tf, the exact relation between parameter pairs and the curves y²=x³−T²x, and the F4+, F7+, and F9+ methods.
Elliptic surfaces for 6/9 patterns
Genus-one quartics, their Jacobians, section ranks, and explicit one-parameter families.
Algebra of magic matrices
The five-dimensional algebra of semimagic squares, its decomposition, and its relation to split quaternions.
Site contents
Sections
Theory
A sequential account from the general form of a magic square to elliptic surfaces.
Contents →Pattern atlases
Orbits of square positions with equations, statuses, and links to derivations.
Computations
Calculators for parametric families, 6/9 patterns, and operations on magic squares.